The biggest issue may be healthcare.
If the mother/child are depending on your healthcare though your employer you need to understand what happens when you turn-in your notice. Some end your coverage at the end of the notice period, some go to the end of the month, others could say today is your last day.
You also need to understand how to new company will cover you. Some cover you on day one, others make you wait to the start of the next month.
Keep in mind that if you move then quit, you might run into an issue if you have to return a laptop, phone, other company equipment.
Or do you think I should work this out with my manager? I do not want
to burn bridges, or at least minimize the damage. But I need the
money, and don't want to have a 2 or 3 week gap in pay either
Normally a person in the US gives notice, and then spends the next two weeks cleaning up projects and preparing to handover information. Your plan to quit, but not be there to help, will burn-bridges. You won't be the first to do this. I have seen it done a couple of times, and nobody is ever happy about it.
Based on a comment you made:
yes child has been born already.. in fact she's 7 months. But I had
not used all my paternity leave I am entitled to (although I wonder if
things would change if I put in notice).
Management generally doesn't like it when a departing employee takes leave during their notice period. It makes it very hard to have a successful turnover. Even if you can get them to approve paternity leave quickly, they may decide that the moment you start your paternity leave you are no longer an employee because you will not be able to help them.